MIMO (multiple input multiple output) refers to a method for improving transmission/reception data efficiency using a plurality of transmit (Tx) antenna and a plurality of receive (Rx) antennas instead of a single Tx antenna and a single Rx antenna. That is, MIMO is a scheme in which a transmitting end or a receiving end of a wireless communication system improves capacity or enhances performance using multiple antennas. MIMO may be referred to as multi-antenna technology.
To support multi-antenna transmission, a precoding matrix for appropriately distributing transmission information to antenna s according to channel state can be applied. 3GPP (3rd Generation Partnership Project) LTE (Long Term Evolution) supports a maximum of 4 Tx antennas (4Tx) for downlink transmission and defines a precoding codebook for downlink transmission using the Tx antennas.
In a MIMO based cellular communication environment, data transfer rate can be improved through beamforming between a transmitting end and a receiving end. Whether beamforming is applied is determined based on channel information, a channel estimated through a reference signal at the receiving end is appropriately quantized using a codebook and fed back to the transmitting end.
A description will be given of a spatial channel matrix (or channel matrix) which can be used to generate a codebook. The spatial channel matrix (or channel matrix) can be represented as follows.
      H    ⁡          (              i        ,        k            )        =      [                                                      h                              1                ,                1                                      ⁡                          (                              i                ,                k                            )                                                                          h                              1                ,                2                                      ⁡                          (                              i                ,                k                            )                                                …                                                    h                              1                ,                Nt                                      ⁡                          (                              i                ,                k                            )                                                                                      h                              2                ,                1                                      ⁡                          (                              i                ,                k                            )                                                                          h                              2                ,                2                                      ⁡                          (                              i                ,                k                            )                                                …                                                    h                              2                ,                Nt                                      ⁡                          (                              i                ,                k                            )                                                            ⋮                          ⋮                          ⋱                          ⋮                                                                h                              Nr                ,                1                                      ⁡                          (                              i                ,                k                            )                                                                          h                              Nr                ,                2                                      ⁡                          (                              i                ,                k                            )                                                …                                                    h                              Nr                ,                Nt                                      ⁡                          (                              i                ,                k                            )                                            ]  
Here, H(i,k) represents a spatial channel matrix, Nr denotes the number of Rx antennas, Nr denotes the number of Tx antennas, r is an Rx antenna index, t is a Tx antenna index, i represents an OFDM (or SC-FDMA) symbol index and k represents a subcarrier index.
hrt(i,k) is an element of the channel matrix H(i,k) and represents an r-th channel state and a t-th antenna corresponding to an i-th symbol and k-th subcarrier.
In addition, a spatial channel covariance matrix which can be used in the present invention will now be briefly described. The spatial channel covariance matrix can be represented by R.
  R  =      E    ⁡          [                        H                      i            ,            k                    H                ⁢                  H                      i            ,            k                              ]      where H denotes a spatial channel matrix and R denotes a spatial channel covariance matrix. In addition, E[] represents the mean, i represents a symbol index and k represents a frequency index.
SVD (Singular Value Decomposition) is a method for decomposing a rectangular matrix, which is widely used in signal processing and statistics. SVD is to normalize matrix spectrum theory for an arbitrary rectangular matrix. An orthogonal square matrix can be decomposed into diagonal matrices using an Eigen value as a basis using spectrum theory. When it is assumed that the channel matrix H is an m×n matrix composed of a plurality of set elements, the matrix H can be represented as a product of three matrices as follows.Hm×n=Um×nΣm×nVn×nH 
Here, U and V represent unitary matrices and Σ denotes an m×n matrix including a non-zero singular value. The singular value is Σ=diag (σ1 . . . σr), σi=√{square root over (λi)}. Representation as a product of three matrices is referred to as SVD. SVD can handle normal matrices, compared to Eigen value decomposition which can decompose only orthogonal square matrices. SVD and Eigen value composition are related to each other.
When the matrix H is a positive definite Hermitian matrix, all Eigen values of H are non-negative real numbers. Here, singular values and singular vectors of H are equal to Eigen values and Eigen vectors of H. Eigen value decomposition (EVD) can be represented as follows (here, Eigen values may be λ1, . . . ,λr).HHH=(UΣVH)(UΣVH)=UΣΣTUH HHH=(UΣVH)H (UΣVH)H=VΣTΣV 
Here, Eigen values can be λ1, . . . , λr. Information on U between U and V, which indicate channel directions, can be known through singular value decomposition of HHH and information on V can be known through singular value decomposition of HHH. In general, a transmitting end and a receiving end respectively perform beamforming in order to achieve higher throughput in multi-user MIMO (MU-MIMO). When a receiving end beam and a transmitting end beam are represented by matrices T and W, a channel to which beamforming is applied is indicated by THW=TU(Σ)VW. Accordingly, it is desirable to generate the receiving end beam on the basis of U and to generate the transmitting end beam on the basis of V in order to accomplish higher throughput.
In designing a codebook, it is necessary to reduce feedback overhead using as few bits as possible and to correctly quantize a channel to obtain a sufficient beamforming gain. One of codebook design schemes presented or adopted as a standard by recent mobile communication systems, such as 3GPP LTE (3rd Generation Partnership Project Long Term Evolution), LTE-Advanced and IEEE 16 m, is to transform a codebook using a long-term covariance matrix of a channel, as represented by Equation 1.W′=norm(RW)  [Equation 1]
Here, W denotes an existing codebook generated to reflect short-term channel information, R denotes a long-term covariance matrix of channel matrix H, norm(A) represents a matrix in which norm is normalized into 1 per column of matrix A, and W′ represents a final codebook generated by transforming the codebook W using the channel matrix H, the long-term covariance matrix R of the channel matrix H and a norm function.
The long-term covariance matrix R of the channel matrix H can be represented as Equation 2.
                    R        =                              E            ⁡                          [                                                H                  H                                ⁢                H                            ]                                =                                    V              ⁢                                                          ⁢              Λ              ⁢                                                          ⁢                              V                H                                      =                                          ∑                                  i                  =                  1                                Nt                            ⁢                                                          ⁢                                                σ                  i                                ⁢                                  v                  i                                ⁢                                  v                  i                  H                                                                                        [                  Equation          ⁢                                          ⁢          2                ]            
Here, the long-term covariance matrix R of the channel matrix H is decomposed into VΛVH according to singular value decomposition. V is an Nt×Nt unitary matrix having Vi as an i-th column vector, Λ is a diagonal matrix having σi as an i-th diagonal component and VH is a Hermitian matrix of V. In addition, σi and Vi respectively denote an i-th singular value and an i-th singular column vector corresponding thereto (σ1≧σ2≧. . . ≧Nt).